Show Maximal Ideal Containment in a PID with ACC

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In summary, in a PID, every ideal is contained in a maximal ideal and this can be proven using the Ascending Chain Condition for Ideals. This result is based on the fact that every ideal in a PID is a principal ideal domain and that if p is an irreducible element of a PID, then <p> is a maximal ideal. The proof involves considering three cases: when the ideal is a field, when the ideal has an irreducible factor, and when the ideal is reducible. The use of the Ascending Chain Condition is necessary for this proof. It is also worth noting that the fact that a nonzero nonunit has an irreducible factor can be proved using the fact that every ideal is contained in a maximal ideal
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samkolb
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Homework Statement


Show that in a PID, every ideal is contained in a maximal ideal.

Hint: Use the Ascending Chain Condition for Ideals


Homework Equations



Every ideal in a PID is a principal ideal domain.

If p is an irreducible element of a PID, then <p> is a maximal ideal.

The Attempt at a Solution



Let D be a PID and let N = <a> be an ideal.

I assumed that N does not equal D, since no maximal ideal of D can contain D.
It follows that a is not a unit in D.

If a=0 and D contains no nonzero nonunit element, then D is a field and <a>=<0> is a maximal ideal. If a=0 and D contains a nonzero nonunit b, then b has an irreducible factor p. So N=<a>=<0> is contained in <p>, which is maximal.

If a is a nonzero nonunit which is itself irreducible, then <a> is a maximal ideal of D.

If a is a nonzero nonunit which is reducible, then a=cq where c is a nonunit and q is irreducible. It follows that N=<a> is contained in <q>, which is maximal.

This seems right to me. I'm writing this up because I never used the Ascending Chain Condition.
 
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  • #2
Twice you used the fact that a nonzero nonunit has an irreducible factor. This fact is often proved using ACC.

You might want to double check that your textbook's proof of "nonzero nonunit has an irreducible factor" didn't use as one of its steps "every ideal is contained in a maximal ideal" and leave the proof as an exercise. :)
 

1. What is a PID with ACC?

A PID (Principal Ideal Domain) is a commutative ring with unity in which every ideal is principal, meaning it is generated by a single element. ACC (Ascending Chain Condition) is a property of PIDs that states every increasing chain of ideals eventually stabilizes, meaning there is a largest ideal in the chain.

2. What is a maximal ideal?

An ideal in a ring is maximal if it is not contained in any other proper ideal. In other words, there is no larger ideal that properly contains the maximal ideal. In a PID, maximal ideals are always prime ideals, meaning they generate prime ideals.

3. What does it mean to show maximal ideal containment?

In a PID with ACC, maximal ideal containment refers to proving that a maximal ideal M is contained in another ideal N, but M is not equal to N. This is an important step in understanding the structure of the ring and its ideals.

4. How do you show maximal ideal containment in a PID with ACC?

To show maximal ideal containment in a PID with ACC, you need to use the fact that every ideal in a PID is generated by a single element. This allows you to compare the generators of the two ideals and show that one is contained in the other. You also need to use the property of ACC to show that the chain of ideals eventually stabilizes.

5. Why is showing maximal ideal containment important?

Showing maximal ideal containment in a PID with ACC helps to determine the structure of the ring and its ideals. This is important in understanding the algebraic properties of the ring and its elements. It also plays a crucial role in other areas of mathematics, such as algebraic number theory and algebraic geometry.

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